Algebraic and Dirac-Hestenes Spinors and Spinor Fields

Abstract Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF ) and Dirac-Hestenes spinor fields (DHSF ) on Minkowski spacetime as some equivalence classes of pairs (Ξu , ψΞu ), where Ξu is a spinorial frame field and ψΞu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the ‘bilinear covariants’ (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary RiemannCartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of
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Introduction
Physicists usually make first contact with Dirac spinors and Dirac spinor fields when they study relativistic quantum theory. At that stage they are supposed to have had contact with a good introduction to relativity theory and know the importance of the Lorentz and Poincar´ e groups. So, they are told that Dirac spinors are elements of a complex 4-dimensional space C4 , which are the carrier space of a particular representation of the Lorentz group. They are told that when you do a Lorentz transformations Dirac spinors behave in a certain way, which is different from the way vectors and tensors behave under the same transformation. Dirac matrices are introduced as certain matrices on C(4) satisfying certain anticommutation rules and it is said that they close a particular Clifford algebra, known as Dirac algebra. The next step is to introduce Dirac wave functions. These are mappings, Ψ : M → C4 , from Minkowski spacetime M (at that stage often introduced as an affine space) to the space C4 . The set of all these mappings is trestricted by imposing to it the structure of a Hilbert space. After that, Dirac equation, which is a first order partial differential equation is introduced for Ψ(x). Physics come into play by interpreting Ψ(x) as the quantum wave function of the electron. Problems with this theory are discussed and it is pointed out that the difficulties can only be solved in relativistic quantum theory, where the Dirac spinor field, gains a new status. It is no more simply a mapping Ψ : M → C4 , but a more complicated object (it becomes an operator valued distribution in a given Hilbert space 1 ) whose expectation values on certain one particle states can be represented by objects like Ψ. From a pragmatic point of view, only this knowledge this is more than satisfactory. However, that approach, we believe, is not a satisfactory one to any scientist with an enquiring mind, in particular to one with is worried with the foundations of quantum theory. For such person the first question which certainly occurs is: what is the geometrical meaning of the Dirac spinor wave function. From where did this concept came from? Pure mathematicians, which study the theory of Clifford algebras, e.g., using Chevalley’s classical books [38, 39] learn that spinors are elements of certain minimal ideals2 in Clifford algebras. In particular Dirac spinors are the elements of a minimal ideal in a particular Clifford algebra, the Dirac algebra. Of, course, the relation of that approach (algebraic spinors), with the one learned
∗ published:
Journal of Mathematical Physics 45(7), 2908-2944 (2004).
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Clifford algebras (including many ‘tricks of the trade’) necessary for the intelligibility of the text.
3 Dirac-Hestenes Spinors (DHS ) 4 Clifford Fields, ASF and DHSF 5 The Dirac-Hestenes Equation (DHE )
6 Justification of the Transformation Laws of DHSF based on the Fiersz Identities. 18 7 Dirac Equation in Terms of ASF 20
arXiv:math-ph/0212030v6 31 May 2005
Algebraic and Dirac-Hestenes Spinors and Spinor Fields ∗
Waldyr A. Rodrigues, Jr. Institute of Mathematics, Statistics e Scientific Computation IMECC-UNICAMP CP 6065 13083-970 Campinas, SP Brazil e-mail:walrod@.br or walrod@ime.unicamp.br last revised: 04/06/2004 (final version)
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C R⋆ 37 p,q , Clifford, Pinor and Spinor Groups C.1 Lie Algebra of Spine . . . . . . . . . . . . . . . . . . . . . . . 39 1,3 D Spinor Representations of R4,1 , R+ 4,1 and R1,3 E What is a Covariant Dirac Spinor (CDS ) 39 42
8 Misunderstandings Concerning Coordinate Representations of the Dirac and Dirac-Hestenes Equations. 21 9 Conclusions 23
A Some Features about Real and Complex Clifford Algebras 24 A.1 Definition of the Clifford Algebra C ℓ(V, b) . . . . . . . . . . . . . 24 A.2 Scalar product of multivectors . . . . . . . . . . . . . . . . . . . . 26 A.3 Interior Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.4 Clifford Algebra C ℓ(V, b) . . . . . . . . . . . . . . . . . . . . . . 27 A.5 Relation Between the Exterior and the Clifford Algebras and the Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A.6 Some Useful Properties of the Real Clifford Algebras C ℓ(V , g) . . 28 B Representation Theory of the Real Clifford Algebras Rp,q 29 B.1 Some Results from the Representation Theory of Associative Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 B.2 Minimal Lateral Ideals of Rp,q . . . . . . . . . . . . . . . . . . . . 35 B.2.1 Algorithm for Finding Primitive Idempotents of Rp,q . . . 36
Contents
1 Introduction 2 Algebraic Spinors 2.1 Geometrical Equivalence of Representation Modules of Simple Clifford Algebras C ℓ(V, g) . . . . . . . . . . . . . . . . . . . . . . 2.2 Algebraic Spinors of Type IΞu . . . . . . . . . . . . . . . . . . . 2.3 Algebraic Dirac Spinors . . . . . . . . . . . . . . . . . . . . . . . 3 7 8 11 12 12 13 16